Answer:
[tex]comp_{\vec{a}}\vec{b}=0.11 [/tex]
[tex]proj_{\vec{a}}\vec{b}=\left ( \frac{4}{81},\frac{7}{81},\frac{-4}{81} \right )[/tex]
Step-by-step explanation:
a=(4,7,-4) b=(3,-1,1)
Scalar projection of b onto a
[tex]comp_{\vec{a}}\vec{b}=\frac{a\cdot b}{|a|}[/tex]
[tex]a\cdot b=\left ( 4\times 3 \right )+\left ( 7\times -1 \right )+\left ( -4\times 1 \right )=1[/tex]
[tex]|a|=\sqrt{4^2+7^2+4^2}=9[/tex]
[tex]comp_{\vec{a}}\vec{b}=\frac{a\cdot b}{|a|}=\frac{1}{9}\\\Rightarrow comp_{\vec{a}}\vec{b}=0.11 [/tex]
Vector projection of b onto a
[tex]proj_{\vec{a}}\vec{b}=\frac{a\cdot b}{|a|^2}\cdot a[/tex]
[tex]\frac{a\cdot b}{|a|}=\frac{1}{9}[/tex]
[tex]\frac{a\cdot b}{|a|^2}\cdot a=\frac{1}{81}\left ( {4},{7},{-4} \right )[/tex]
[tex]proj_{\vec{a}}\vec{b}=\left ( \frac{4}{81},\frac{7}{81},\frac{-4}{81} \right )[/tex]
